Let me start by saying what is written in grammar books on this issue and after that I will put my question. (Take heed that this usage of "could" ONLY refers to the past and ONLY to affirmative sentences)

1) "Could" shouldn't be used to address a single action/achievement in the past but it can be used to address a general ability in the past.

The examples are from M. Swan's grammar.

a) When I was younger, I could run 10 km in under 40 minutes. (CORRECT)
b) I could run 10 km in under 1 hour yesterday. (WRONG)

It's all been clear so far. But here are a few answers of native speakers which I have got trying to elaborate.

Examples and opinions of native speakers on which is correct/incorrect.
*1) First native speaker's examples*
"I could visit you yesterday" - (incorrect)
"I could do it yesterday" - (correct)
"I could see you yesterday" - (correct)

My thoughts
I don't quite understand why "I could visit" is incorrect while "I could do/see" is correct. Does it have to do something with single/multiple action?

That is,

"I could visit you (only ONE TIME) yesterday" if (only ONE TIME) then it's (incorrect)
"I could do it (MANY TIMES) yesterday" if (MANY TIMES) then it's (correct)
"I could see you (MANY TIMES) yesterday" if (MANY TIMES) then it's (correct)

2) 1) Second native speaker's examples

"I could visit you yesterday, but I didn't want to" - (correct)
"I could kill everyone at work last week, but I couldn't have done it." - (correct)

My thoughts

I am confused as for why in this case "I could visit you yesterday" is considered acceptable. Does it mean that: "I could visit you yesterday" may have more than one meaning? Does it mean that "I could run 10 km yesterday" can also mean something and be correct?

Plus one more question. What does it mean? Does it refer to the future or the past

I can't agree with the "correct/incorrect" judgments about all those sentences. Native speakers don't always know what they're talking about. We all make mistakes. Swann's sentences & his "rule" seem fine. The 1st native speaker is confused, inconsistent, & contradictory. The 2nd native speaker is illiterate & uneducated. I think the question should be reorganized & rewritten to ask whether the 5 example sentences are OK & why or why not. I'd edit it for you, but that would be too radical an edit. There are too many thoughts and too many questions here.
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user21497Feb 19 '13 at 14:29

Everything depends on how you framed your questions to your informants. Did you do so in a way which either explicitly or contextually distinguished between indicative and conditional use?
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StoneyBFeb 19 '13 at 16:34

1

Is this a question about the difference in usage and/or meaning of "could" vs using "to be able?"
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livresqueFeb 19 '13 at 19:51

I like your response to the first half of OP's question, but your response to OP's 2nd question is not correct. It clearly refers to a past context, so those sentences should also use could have.: I could have visited... and I could have killed everyone.... The alternate sentences you have provided have completely different meanings, temporally or otherwise.
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Shawn MooneyFeb 19 '13 at 14:43

@ShawnMooney You really think that “You could ask me before calling him” is actually referring to events in the past? I certainly do not.
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tchristFeb 19 '13 at 14:58

1

No I do not think that. What I have said (or tried to say) is that OP's #2 question and example sentences should grammatically without-a-doubt be located in the past, and thus are ungrammatical; furthermore, that your suggested revisions thereto are ungrammatical vis-a-vis what OP was inquiring about. In other words, your answer to #2 of OP is unhelpful because you have shifted tenses--to be honest, wildly--from what OP was inquiring about.
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Shawn MooneyFeb 19 '13 at 15:08

1

@ShawnMooney "I could not visit you yesterday" would refer to the past and be grammatically correct. Let's throw a wrench in the VP.
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livresqueFeb 19 '13 at 19:42

The use of could in terms of "I could run 10km in under 40minutes" refers to a general state that holds for some time, rather than a temporary one, and generally implies that the possibility was indeed exercised some of the time.

As a child, I could visit my grandmother whenever I wanted, as she lived only two doors away.

We would assume that not only was it possible for the author to have done so as a child, but they did.

In contrast, could have refers to a possibility that didn't happen.

I could have visited my grandmother when I was in my home-town, yesterday.

We'd assume that the author did not make such a visit.

*I could visit you yesterday.

Suggests that over the course of yesterday, you could visit the addressee, and indeed this did come to pass several times. That's not the likely intended meaning, and nor is the direct meaning likely enough for us to take it as meaning that; it's so strange as to read as an outright grammar error.

@AndrewLeach as do I, clearly my typo was a case of parapraxis. (Though more seriously, since the source of writing mistakes is close to our general topic, I suspect I was subconsciously trying to avoid ambiguity with exercise in the sense of the physical exercise that running is, and that led me away from the correct term).
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Jon HannaFeb 19 '13 at 14:37

Jon, clarify please! Am I right on the following? If we add to this sentence "I could run 10km in under 40minutes" the word "yesterday" then it becomes incorrect? That is, "I could run 10km in under 40minutes" - CORRECT (some time in the past) / "I could run 10km in under 40minutes yesterday" - incorrect.
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user1425Feb 19 '13 at 15:55

2

It's correct if you were in a horrible accident, or fell sick to a dreadful wasting disease. Otherwise if you could do so yesterday then you should surely be able to today: It's not ungrammatical, it's just a strange thing to say in most circumstances, which makes it sound strange.
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Jon HannaFeb 19 '13 at 16:10

3

I agree with Jon Hanna: "I could visit you yesterday, but the bridge was washed out by a flood last night, so I can't today."
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Peter Shor Feb 19 '13 at 16:25

@PeterShor As far as the title of "could" vs "to be able," which I feel somehow got lost in the question, how about the meaning and usage of "I could visit you yesterday" versus "I was able to visit you yesterday?"
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livresqueFeb 19 '13 at 19:47